Four Dimensional Real Time Kinematic State Estimation and Analysis of Relative Clock Solutions

Transcription

1 Four Diensional Real Tie Kineatic State Estiation and Analysis o Relative Cloc Solutions Yaning Feng Queensland University o Technology, Australia Boeng Li Tongi University, P.R. China BIOGRAPHY Yaning Feng is a Proessor o Queensland University o Technology in Brisbane Australia where he is responsible or a research group or wireless counications and GNSS navigation. In recent years, he has been a proect leader or a nuber o Australian Cooperative Research Centre (CRC) unded research progras including Multi-GNSS data processing strategies and services and Wireless Counication Standard Fraewor. His active research interests include satellite orbit deterination, real tie ineatic positioning and tiing, data processing o ultiple GNSS requency signals and Dedicated Short-Range Counications (DSRC) or road saety. He is the Editor-in-Chie or Journal o Global Positioning Systes. He received PhD degree in Satellite Geodesy ro Wuhan Technical University o Surveying and Mapping (now part o Wuhan University), China in 99. Boeng Li received his BSc. and PhD degree in Geodesy and Surveying Engineering ro Tongi University, China, in 5 and respectively. He is now a Research Fellow in GNSS research centre, Curtin University o Technology, Australia. His research interests include integer estiation and GNSS abiguity resolution, Networ RTK, Data processing theory o ultiple requency GNSS signals, Geodetic data processing theory etc. Dr. Li has published ore than 4 peer-reviewed papers in GNSS and Geodetic data processing theory, and received ore than awards ro dierent organizations including the ION GNSS 8 sponsored student paper. ABSTRACT GNSS counity has developed a series o GPS-based techniques or cloc easureents and coparisons including tie disseination with the standard GPS point positioning ethod, cloc synchronization using the codebased dierential GPS positioning technique and using precise point positioning (PPP) or cloc coparison. In the ean tie, signiicant advances have been ade towards real tie ineatic (RTK) positioning which enables three diensional (3D) positioning at centieter accuracy, but without cloc solutions. This paper presents a our diensional real tie ineatic (4D-RTK) ethod, which provides RTK tiing solutions or applications such as cloc synchronization and reote clocs steering in addition to 3D RTK position solutions as usual. The proposed 4D RTK odel is based on n single dierenced (SD) easureents between receivers or a baseline with (n-) double dierenced (DD) integer constraints. The existing 3D RTK odel use (n-) DD code and phase easureents can only enable position estiation, eliinating the receiver cloc errors. In the 4D RTK, the user 3D position states are deterined as usual with the DD phase easureents whose integer abiguities are searched and ixed using the existing integer least squares (ILS) procedure. Estiation o the relative cloc bias and SD phase biases is based on the SD code and phase observational equations. The constant nature o the SD phase abiguity biases is used to iprove the SD phase bias ro epoch to epoch, and thus the relative cloc solutions. The 4D RTK can be sipliied to a RTK tiing proble i the user baselines are given. Experiental analysis has been perored with a GPS data set collected over a baseline at 5 second interval or a total o 85 inutes. The results show that within the irst ew inutes o observations, the SD phase biases all within the range o.3 cycles, thus resulting in the cloc uncertainty o. to. ns. Beyond 5 epochs (about 4 inutes), the SD phase biases are stabilized within the range o. cycle, thus the cloc biases can be estiated to the precision and accuracy o better than. ns, including the eect o the errors in the SD phase easureents. In general RTK tiing has a signiicant potential or high precision real tie cloc synchronizations over very long distances.. INTRODUCTION With suicient nuber o Global Navigation Satellite Syste (GNSS) code and phase easureents, user state paraeters, position, velocity and tie (PVT), can be estiated at the dierent accuracy levels. Signiicant advances have been ade towards the iproveent o real tie position estiation to accuracy down to the centieter level using techniques such as precise point positioning (PPP) and real tie ineatic (RTK) 3rd International Technical Meeting o the Satellite Division o The Institute o Navigation, Portland, OR, Septeber -4, 9

2 positioning in the past two decades. GNSS counity has developed a series o GPS-based techniques or cloc easureents and coparisons since early 98s, including the contributions by Allan et al (98), Lewandowsi et al (99) and Miranian et al (99). The siplest GNSS tie transer ethod is GPS being used as a cloc in the one-way ode (OWM) where a single GPS receiver is anywhere to provide cloc solutions at the precision and accuracy o -3 ns since selective availability turned to zero in. Parer et al (4) outlined the advances in tie and requency disseination using GPS techniques. The second ethod is to use the dierential GPS to synchronize two clocs separated by large distances in the coon-view ode to the precision and accuracy o 5- ns. Schildnecht () exained the high precision tie and requency transer using GPS phase easureents. Another way to synchronize clocs is the elting-pot ethod (MPM), which is siilar to both OWM and CVM ethods, but allows clocs at reote sites to be autoatically controlled and steered. The latest technique is PPP, a technique initially developed or deterining positions with sub-decieter accuracy ro single-receiver easureents. PPP oers high-level perorance coparable with state-o-the-art ethods, such as Two-Way Satellite Tie and Frequency Transer (TWSTFT), autonoously allowing recovery o the IGS cobined cloc solution at sub-nanosecond level (Orgiazzi et al). In general, i PPP technique can provide positioning solutions o centieters in real tie, it is possible provide real tie cloc solutions to the RMS accuracy o. ~.3 ns with an advanced data processing syste and the RMS is o.3 ns as shown in (ibid). Using precise GPS orbit and clocs or precise user cloc estiations has attracted a signiicant research attention within the International GNSS Services (IGS) counity, reerring to the contributions by Kouba et al () and Ray et al (, 3). The advantages o PPP tie solutions is that the user can directly obtain the tie solutions in the UTC rae, considering the leap seconds between GPS tie and UTC tie. The disadvantages o PPP include long convergence tie or desirable accuracy and higher latency o solutions with respect to RTK solutions. However, currently, RTK solutions are liited to 3D coordinates. This paper presents a our-diensional real tie ineatic (4D-RTK) positioning and tiing ethod, which provides precise relative cloc solutions or applications such as synchronization and autoation o reote clocs steering. The paper presents odels and algoriths or 4D-RTK tiing and experiental results. 4D RTK is a relative 3D positioning plus D tiing concept. In Section, the rigorous 4D RTK odels and algoriths are given, which are based on n betweenreceiver single dierenced (SD) easureents or a baseline with (n-) double dierenced (DD) integer constraints. Section 3 exaines the eects o interrequency and inter-receiver biases on the relative cloc solutions. Section 4 gives experiental analysis with a data set o baseline, showing that within a ew inutes o observations, the relative cloc bias can coverage to the accuracy o a ew centieters, that is,.~. ns. The inal section concludes the paper with a suary o indings in this contribution.. 4D RTK odels and algoriths The 4D RTK is based on the between-receiver SD phase and code easureents. For siplicity, the linear equations or SD phase easureents are expressed as L, a, N ε L, δx cδt λ M M + + M + M () L n, a n, N n ε Ln, where there are n satellites in view at the th epoch; L i is the ith SD phase easureent or an original phase signal L, or a cobined signal ro L and L, which is containated by rando error ε L i and inter-requencybias, and a i is a 3 cosine vector or direction ro satellite to receiver; δx, δt and N i are 3D coordinate unnowns, SD cloc error and the ith SD abiguity with initial phase bias which are presented in real values; c is the light speed and λ is the wavelength o the carrier phase signal L i at its requency. The subscript denotes the th epoch. Siilarly, the linear equations or code easureents is given P, a ε, P, M M δx + cδt + M () P n, a n, ε Pn, where P i, is the ith SD code easureent or the P (or C/A) code, or cobined easureent ro P and P as shown in Table ; ε is its rando error. The reaining Pi variables have the sae eanings as those in equation (). The equations () and () or phase and code easureents are obtained ro standard SD processing between receivers, where () is the typical code based dierential GPS odel. In 4D RTK processing, the position and receiver cloc paraeters vary ro epoch to epoch without dependences on cloc odels, but the SD abiguity reains stable and generally treated as constant over a certain tie interval. The SD observation equations at the th epoch in atrix or reads L % % δz +, σ σ Q y A B εy Q (3) N QP where 3rd International Technical Meeting o the Satellite Division o The Institute o Navigation, Portland, OR, Septeber -4, 93

3 y L, L Ln, P, L P n,, N a, δ X δz δt, N M, A M N n a n, %A [ ] A e, % λin B B λi T, n where I n is identity atrix with n diension; σ is the variance o unit weight or SD phase observation and Q L is its coactor atrix as o the covariance atrix σ. Q L The coactor atrix Q P o SD pseudoranges is satisied with QP σp σ Q L,where σ P is the variance o the SD pseudorange noise. It is iportant to notice that we use δ T in eter to denote cδ T. The LS solution o the SD observation equation syste (3) is T T T δz AQ % A% AQ % B% AQ % y T T T (4) N BQ % A% % % BQ B BQ % y In general, easureents ro ultiple epochs would be cuulated or the SD abiguity solutions. For siplicity, δz is equivalently eliinated or each epoch and only SD abiguity paraeters are retained, i.e., Ry RBN % + ε y (5) where the transoration atrix T T R I A % A % Q A % A % Q ( ) which is an idepotent atrix satisying with T T RR R, Q R RQ and Q R RQ. With easureents cuulated over the ost current epochs, the noral equations associated to the observation equation (5) are Q N w (6a) N where the coactor atrix o SD abiguities is T Q ( ) B% Q % N R B (6b) and the constant ter T w B% Q R y (6c) Thus the LS solution is N QN w (6d) To deal with the situation where a satellite is added or reoved, we arrange the observations into two groups. At the current epoch, the atrices Q, B % and y, are expressed as ollows Q B% y Q,, B% y Q B % y The subscript represents the group satellites which are coon to two consecutive epochs, and the subscript the group satellites which are either added or reoved at the current epoch. In the case when the group satellites are added, the su o the atrices in (6b) is expressed as R R R + R R and the su o the vectors becoes in (6c) is Ry R R y + R R y In the case when the group satellites are reoved, the su o the atrices in (6b) will becoe ( R ) + [ R ] where the irst ter represents the sub-atrix including the coluns and rows related to the group satellites only. Siilarly, the su o the vectors in (6c) is given as ( R ) y + [ Ry ] where the irst ter represents the sub-vector including coponents only related to the group satellites. It is iportant to note that SD abiguities N are realvalued due to the eects o initial phase biases and other receiver-speciic biases, such as initial phase errors, but their dierences aongst the satellites are integers in theory. To achieve ore precise SD abiguity and thus user state solutions, these integer constraints should be applied. Assuing the irst satellite to be the reerence satellite, we derive the DD loat abiguity solution by ultiplying the DD operation atrix D [ en I n] to SD abiguity solution, naely, the LS DD loat solution is Δ N D N (7a) and its coactor atrix is T QΔ DQ N ND (7b) The DD abiguities ΔN is then deterined by solving the ollowing iniization proble Δ N ( arg in ( ΔNΔN T ) Q ( ΔNΔN ) (8) n ΔN ΔN The search space is governed by the covariance atrix σ Q ΔN and the strong correlation within Q ΔN can lead to larger search space. With the well-established decorrelation technique such as LAMBDA, the integer search can be copleted quicly. Once the ixed integer solutions Δ N ( are obtained, we can ollow two approaches to derive the 3D and position and relative cloc solutions. The irst approach is to ipose 3rd International Technical Meeting o the Satellite Division o The Institute o Navigation, Portland, OR, Septeber -4, 94

4 the constraints D N Δ N ( upon observation equations (5), the noral equations (6a) are updated to include these constraints as T Q D NC w N ( Δ (9) D N Thus the solution o SD abiguities with DD integer constraints becoes T ( ) T N C In Q D QΔD N+ Q D QΔΔ N ( (a) N N N N and its coactor atrix T T Q ( IQ D Q D) Q ( ID Q N DQ ) C N ΔN N ΔN N (b) Soe coents ollow ro (a) and (b) or SD abiguity estiation. In principle the accuracy o SD abiguity solution can be iproved with the constraints o ixed DD integer abiguities. However, to which degree the SD abiguity can be iproved with respect to its unconstrained estiate depends on the noise level o the SD abiguity estiates. Substituting the SD abiguity solution with constraints o DD integer abiguities into the noral equations o the latest one epoch siilar to (4), we copute the 4D paraeters at the th epoch T T T δz A% Q A% A% Q y AQ % BN % (a) ( ) ( ) C and their coactor atrix T T T T Q ( ) ( )( ) % % % % + % % δz A Q A A Q A CQ C A N Q A (b) where the ediu variable T T C A% Q B% ( In QND QΔ ND) Q N (c) The second approach is to estiate 3D position states with the ixed DD abiguities, then to deterine the cloc paraeters with the nown position states. It can be proved that the 4D solutions coputed by equation (a) are equivalent to those coputed in the traditional 3D RTK odel using DD easureents with ixed DD abiguities. This equivalence allows sipliication o the algoriths. We can alternatively realize the 4D RTK procedure based on the existing unctions o 3D RTK. The DD linear odel is T Hδ X U, Q DD DQD () where ( DA DL + λδn H, U. DA DP Ater DD abiguities are ixed by Eq.(8) in above procedure, we solve the coordinates based on the noral equations o DD odel () are T T δ X HQ DD H HQ DD U (3) Substituting ( ) δ X into equation (3) yields δt y% A% B% + εy (4) NC where A y% y δ X and %A A e. It is noticed that the SD abiguities in Eq.() have constrained by integer DD abiguities because the coordinates with DD integer abiguities have been applied. The noral equations or the th epoch are obtained T T T AQ % A% AQ % B% δt AQ % y% T T T % % % % C % (5) BQ A BQ B N BQ y% The equivalent noral equations or SD abiguities N is derived T T BQ % RBN % C BQ % Ry % (6)) Siilarly, we solve the precise SD abiguities using the ost recent epochs T T N B% Q R B% B% Q R y % (7) ( ) C Once the SD abiguity N C is solved, we substitute it into equation (5) to copute the cloc solution at the current epoch: T T δt ( A% Q A% ) A% Q ( y% BN % c ) (8) Reerring to the discussions ollowed ro (6b) and (6c), coputations (6) to (8) can consider also adding and reoving satellites. Considering % e A e and siply assuing Q L I n and QP σ P σ QL βi n, where β σ P / σ, we can derive the siple orulae or this case β n δt ( L, i + P, i λnc) i n β + β (9) In the case when the user 3D position states are nown, i.e., the solutions with and without the constraints o DD integers are consistent. For siplicity purpose, the constraints o DD integers are not considered, then the coactor atrix (6b) and constant ter (6c) o SD abiguities becoe β et Q In + N λ n (a) T L P w λ [L ee ( ) /( )] + + n nβ β (b) The LS estiate o the SD abiguity paraeters is given as N Q w (c) N 3rd International Technical Meeting o the Satellite Division o The Institute o Navigation, Portland, OR, Septeber -4, 95

5 O course, we can ix the DD abiguities by Eq.(8) and copute the SD abiguities with DD integer abiguity constraints by Eq.(a) and (b). In general β is signiicantly larger than n, thus the variance or each eleent o N in Eq(a) can be approxiated as σp σ λ Q (b) n It is noted that in the oration o the LS equations (6a)- (6d) and (7)-(b), the possible teporal correlation o SD code and phase easureent errors has not been taen into consideration. We also observe ro (9) that although theoretically all SD abiguities N C ay be estiated ro tie to tie, larger uncertainties in the new rising satellites ay aect the accuracy o the cloc solutions. This suggests that the estiation o the cloc biases ay use phase and code easureents ro selected satellites instead o all. N (i) Fro the above derivations one can easily conclude that with nown baselines or user stations, the RTK cloc solutions can be deterined directly ro the SD phase and code easureents. The DD phase integer abiguity resolution is not required unless very high accuracy is expectable. This sipliies the 4D RTK to RTK tiing. Precise tie synchronization over large distance is ade easy as a long as the ionospheric and tropospheric biases in SD phase easureents are adequately corrected. 3. Corrections or inter-requency biases in relative cloc solutions The hardware and signal processing architecture o a GNSS receiver will introduce requency-dependent biases in the pseudorange and carrier phase easureents. These variations in the easureents, which ay be dierent or code and phase, are coonly nown as interrequency biases (Petovello et al., ). These biases are classiied into two types: satellite-speciic and receiver speciic biases. A one-way pseudo-range or phase GPS easureent is deined as the pseudo-range or phase easureent or one receiver-to-satellite pair, which ay also be called a line-o-sight (LOS) easureent or zero dierence easureent. In the dual-requency GPS case, our easureent equations are written as ollows p p ~ + εp (a) ~ ρ + αi + αtg + i + εp (b) ρ ~ Ii + Tg + λ + ε,i ρ + I + Tg,i R,i b (c),i ρ ~ αii + αtg + λ b + R i + ε (d) where p, p,i,i,i,i are the pseudoranges and phase easureents on L and L carriers respectively. The superscript and subscript i, represent the th satellite and the ith receiver; ρ ~ is the pseudorange, consisting o the geoetric distance between the receiver and satellite, and receiver and satellite cloc biases; λ b and λ b are the phase abiguities in the carrier phase observables on L and L respectively, which are non-integers in the oneway phase, where λ and λ are the respective wavelengths or L and L carriers; T g is the transitter speciic interrequency bias in the code easureents on L or the th satellite; αt g is the transitter speciic inter-requency bias in the code easureents on L or the th satellite, where α ( / ) (77/6) ; R i is the receiver dierential inter-requency bias on L or the ith receiver. Because the tiing o the GPS receives depends on the L C/A code, the inter-requency bias on L is by deinition zero; ε p, ε p are the pseudo-range noises or p and p respectively. The noise ters in this context include eects o ultipath and receiver noise; ε, ε are the carrier phase noises on L and L respectively. The noise ters in this context include eects o ultipath and receiver noises. The eects o the inter-requency biases in the SD pseudorange and phase equations () and () are not explicitly shown up, but they indeed aect the relative cloc results o interest, depending the easureents used. Table suarizes the bias ters in dierent phase and code cobinations in ZD and SD easureents. In DD easureents, all the receiver-speciic and satellitespeciic biases are cancelled. It is observed that in the ionosphere-ree code and phase easureents, the satellite-speciic bias T g will be copletely deleted, but the receiver-speciic bias exists in both ionosphere-ree SD code and phase easureents. Because the receiverspeciic biases are generally stable over tie, the biases in SD code and phase easureents can be pre-calibrated or a particular receiver-pair using zero or very shortbaselines on regular basis. 4 Experients and Analysis We use one GPS data set to deonstrate the perorance o the relative cloc solutions with the proposed 4D RTK ethod. Table suarizes the inoration about the data and settings in the 4D RTK data processing. Figure (a) plots the 8 L SD phase abiguity solutions (ractional) obtained using L and P code easureents and ollowing the equations (7) and (8). It is clearly seen that within the ew inutes o observations, the SD phase biases all within.3 cycles; beyond 5th epoch (about 38 in or 5s siple intervals), each SD solution is stabilized within. cycle. Figure (b) plots the sae SD 3rd International Technical Meeting o the Satellite Division o The Institute o Navigation, Portland, OR, Septeber -4, 96

6 phase biases solutions obtained using L phase and the cobined P code easureents. The section (b) shows the convergence o the solutions siilar to (a), but there are a dierence between the P and P-based solutions in both integer and ractional coponents, presuably due to the receiver inter-requency biases (IFB), which reains constant over hours to days, can possibly be corrected beorehand according to the Table. Table : Inluence o inter-requency biases on the ZD, SD and DD Cobination Bias ter in ZD Bias in SD T c g R Tg R ΔR ΔR p Tg p + p p R ΔR + Tg p p pc R ΔR Note: ΔR is the receiver-speciic bias with respect to the base receiver. Table. Experiental data and setting in the 4D RTK data processing Data Set and setting Source o data S/cors-data.htl, RINEX data: Sites: P474-P478 on //7 Observation types L, L, P,P Distance Cuto elevation 5 degrees Nuber o SVs 8 Data period :9:3~:36: Saple rate 5 seconds Total o epochs 347 Code noise P,P.363 (estiated) (SD) Phase noise L,L.c (estiated) (SD) Figure (a) plots the RTK cloc solutions against the DGPS cloc solutions using the P code easureents, with the dierences as shown in Figure (b). The eects o the receiver dependent inter-requency biases on the relative cloc solutions depend on the cobination o the code easureents. Figure 3 (a) plots the RTK clocs derived ro P and P code easureents respectively, and the dierence between two RTK clocs are shown in Figure 3 (b), which tends to converge when the data is cuulating.. L phase bias(cycle) L phase bias(cycle) (a) Epoch (5 seconds) (b) Epoch (5 seconds) Figure. Illustration o the L SD phase bias solutions (ractional) obtained using L and P code easureents in section (a) and using L and P codes in section (b) respectively. The results ollow ro the equations () and (). It is clearly seen that within the irst inutes, the variation o SD estiates are conined.3 cycles; beyond 5th epoch (about 38 in or 5 siple intervals), each SD solution is then stabilized within. cycle. RTK-DGPS cloc (etre) Relative cloc solutions (etre) (a) Epoch (5 seconds) RTK cloc DGPS cloc (b) Epoch (5 seconds) Figure. Coparison between RTK and DGPS cloc solutions using P code easureents in section (a) and dierences are shown in section (b). As shown in the observation equation (), the accuracy o the RTK cloc solutions is doinated by two actors: () the overall SD phase noise level, which ay include the eects o ionosphere, tropospheric and rando noise and ultipath and () the uncertainty o the SD phase bias solutions being iproved ro tie to tie. In Figure 4 3rd International Technical Meeting o the Satellite Division o The Institute o Navigation, Portland, OR, Septeber -4, 97

7 (a), we show the code STD variations and the overall STD o code P and P are.363 and.63 respectively. The overall noise o the L SD phase observations is estiated to about. c considering both ionospheric and tropospheric eects. As a result, the accuracy o the RTK cloc solutions are plotted in Figure 4 (b), showing the RTK cloc accuracy convergence to the level o.5 c beyond the 5th epochs RTK cloc dierence (etre) RTK cloc solutions (etre) (a) Epoch (5 seconds) RTK cloc with L, P RTK cloc with L, P (b) Epoch (5 seconds) Figure 3. Coparison between two RTK cloc solutions obtained with (L,P) and (L,P) respectively, showing the eects o the receiver-requency related biases. STD o SD Pseudorange() P code: overall STD.363 P code: overall STD (a) Epoch (5 seconds) The proposed 4D RTK odel has been based on SD easureents between receivers. While the user 3D position states are deterined as usual with the DD phase easureents whose integer abiguities are searched and ixed using the existing ILS procedure, the relative cloc bias and the SD phase abiguities are estiated siultaneously epoch by epoch with the SD code and phase observations. The constant nature o the SD phase abiguity paraeters allows their estiations to be iproved accuulatively; in turn the relative cloc solutions are iproved ro tie to tie without cloc odeling. When the user baselines or user coordinates are accurately nown, the RTK cloc solutions can be deterined directly ro the SD phase and code easureents. As the DD phase integer abiguity resolution is no longer required, 4D RTK is reduced to RTK tiing. Precise tie synchronization over large distance thus is thus ade easy. Experiental analysis perored with a GPS data set collected over a baseline at 5 seconds have shown that within the irst inutes o observations, the SD phase biases all within the range o.3 cycles, achieving a. ns cloc uncertainty. Beyond 5 epochs, the SD phase biases are stabilized within the range o +/-.5 c, thus leading the cloc biases estiation to the precision and accuracy o better than. ns, which have included the eect o the errors in the SD phase easureents. ACKNOWLEDGEMENTS STD o RTK clocs () With P code STD (.363) With P code STD (.63) (b) Epoch (5 seconds) This wor was carried out under the partial inancial support o the Australian Cooperative Research Centre or Spatial Inoration proect. The contribution ro the second author was ade during his PhD candidature at Tongi University in China in early. Figure 4 (a) Illustrations o the code STD variations and the overall STD o code P and P are.363 and.63 respectively. The accuracy o the RTK cloc solutions reaches the level o 3 c within a ew inutes, and converges to.5 c beyond the 5th epoch. 5. CONCLUSIONS This paper has developed the 4 D RTK positioning and tiing odels and algoriths that can generate the relative cloc solutions along with 3D position states. The existing 3D RTK odel use DD code and phase easureents is a equivalently reduced case o the 4D RTK proble, but eliinating the receiver cloc errors. REFERENCES Allan, D W. and Weiss, M. A. (98) Accurate Tie and Frequency Transer During the Coon View o a GPS Satellite, Proceedings o 34th Annul Frequency Control Syposiu, 98,p Bauch A., Achar J., Bize S., Calonico D., Dach R., Hlavac R., Lorini L., Parer T., Petit G., Piester D., Szyaniec K., and Uhrich P. (6) Coparison between requency standards in Europe and USA at the 5 Uncertainty Level, Metrologia, 6,43:9. 3rd International Technical Meeting o the Satellite Division o The Institute o Navigation, Portland, OR, Septeber -4, 98

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